Fractions and Irrational Numbers
Fractions Dozenal fractions are written on the same principle as decimal fractions, but the fact that they cover the most important fractions in use - including thirds, which in decimal exhibit infinitely recurring digits - means they are of utility for all walks of life, while relegating vulgar fractions (numerator over denominator, as in '1/3') to infrequent use, where they would mainly be used for fifths, which are problematic when written as dozenal fractions. It has become customary among dozenalists to use the semicolon as the dozenal point, and to refer to it by the name 'dit', although those are only extant practices. Thus a dozenth (twelfth) is written as 0;1 and read '(oh) dit one'. Basic Fractions Fundamental among dozenal fractions are the single-digit fractions, from one to eleven parts in a dozen; the advantage of dozenal is clear in them, for they cover the half, the quarters and the thirds, all of which are needed in practical applications. (In the fraction tables, brackets surround the recurring digits in those fractions that do not terminate in a base.) The ramifications of terminating sixths and thirds in dozenal go beyond simplicity: in a dozenal civilisation there would be little to none of the dichotomy seen in decimal usage between the prevalence of vulgar fractions in everyday life and that of decimal fractions in industrial and scientific settings. Dozenal fractions would be nearly universally used. In addition, dozenal fractions would obviate the need for an auxiliary superbase such as is used for degrees, minutes and seconds in angular measurement in the decimal world; the main reason why radians are avoided in favour of degrees is not because they are based on π (which is simply written out), but because thirds and sixths of π are often needed in practical geometry, but they cannot be written comfortably using decimal fractions. With dozenal fractions, 0;4π radians would be as common in practical geometry and everyday life as it would be in high mathematics. Two-Place Fractions The unreduced eighths, ninths and dozen-fourths ('sixteenths' in decimal nomenclature) take two fractional places in dozenal fractions, as 8, 9 and 14 divide 100 but not 10. Of these, ninths would be most useful, and eights are less important, and dozen-fourths are even less importancess. Here is a table of all the binary fractions down to the fourth binary power: Because 10 has two binary powers in its prime factorisation (2·2·3) as opposed to only one in X (2·5), dozenal fractions need an additional place for every two binary subdivisions added, while decimal fractions add a place for each single binary subdivision. Therefore, while not as indefinitely scalable as in binary bases such as octal and unquadral (which come at the price of covering no other prime factors, and hence having no non-divisor regular numbers), dozenal provides quite well for dealing with binary powers, with only two fractional places needed for all the binary subdivisions in common use (as well as divisibility testing for five binary powers). Non-Terminating Fractions Fractions with a prime factor higher than 3 in their denominator have no exact representation in dozenal. Digits recur infinitely in them; the length of the recurring sequence depends on the relationship of the base (dozen) to the prime factor in question. For example, elevenths have a minimal recurrence of a single digit, because 10 is En−1 (one less than the base, also called the omega relationship); 1/E is 0;(1), with a single recurring digit '1'. In contrast, fifths are maximally recurrent (as 10 is 5n±2), with four digits in the recurring sequence; and so are sevenths, maximally recurrent in dozenal (as 10 is 7n−2) just as they are in decimal (as X is 7n+3), with six digits in the recurring sequence. Here are the basic decimal fractions as they appear in dozenal: One notices not only the recurrence, but also the cycling, of the digits '2497' throughout the series; this is familiar to decimalists in the way the digits '142857' cycle in the sevenths (in dozenal it is the digits '186X35' that cycle in the sevenths). Sevenths are a problem. They are important in dividing weeks. In a dozenal world they would be avoided wherever they could be, just as decimalists try to avoid thirds if possible, only that sevenths would be far easier to avoid than thirds, as they are less important. Where, however, sevenths would need to be represented exactly and could not be avoided, a number of creative solutions might be employed: * Vulgar fractions, writing '1/7' just as decimalists write '1/3' or '1/7' to avoid the recurring digits. Sevenths might well be the one application of vulgar fractions in a dozenal civilisation. * Abbreviated notation, such as using the letters A, B, C, D, E, F to stand for infinitely recurring '186X35', '35186X', '5186X3', '6X3518', '86X351' and 'X35186' (0;A = 0;186X35186X35...). * Auxiliary superbases for niche applications, for example E80, where both the basic dozenal fractions and sevenths are provided for. * Approximation of sevenths using the multiples of 187 (the number that, when multiplied by 7, produces 1001, one more than the cube of the base): 000, 187, 352, 519, 6X4, 86E and X36. Precision can be added to the approximations at will, so that 1/7 would be 0;187 or 0;186E or 0;186X4 at the discretion of the user (analogous to approximating 1/3 as 0.333 or 0.3333 or 0.33333 and so on in decimal). Apart from the sevenths, which would only occasionally be needed, the non-terminating fractions of dozenal would be nothing more than curiosities, in contrast to the way our decimal world is beset by the recurrent digits of thirds, sixths and the rest. Irrational Numbers An irrational number cannot be expressed as a ratio of two integers; hence, they have no terminating representation in any base. At best, they may have a closeness to a fraction of the base that lends itself to easy memorisation. That is a matter of chance, of course, but dozenal has quite a few useful irrational numbers with short approximations. The most common irrational numbers are given in the following table. Note again how dozenal fractions enable us to express cube roots without having to use the special symbol; 'n raised to the power of a third' is readily written with a dozenal fraction. The dozenal representations are given to 10 fractional places. Of these, the square root of 2 and the golden ratio (φ) have exceptionally succinct dozenal approximations, and the precision of π may be set as 3;1848, 3;18480949 or 3;184809494. Category:Pages